Grassmann Algebra

ExploringGrassmannAlgebra.nb 11
� The interior product of Grassmann numbers
The interior product of two elements Α and Β where m is less than k, is always zero. Thus the
m k
interior product X ���� Y is zero if the components of X are of lesser grade than those of Y. For two
general Grassmann numbers we will therefore have some of the products of the components
being zero.
By way of example we take two general Grassmann numbers U and V in 2-space:
�2; �U � CreateGrassmannNumber�Υ�,
V � CreateGrassmannNumber���
�Υ0 � e1 Υ1 � e2 Υ2 �Υ3 e1 � e2, Ω0 � e1 Ω1 � e2 Ω2 �Ω3 e1 � e2�
The interior product of U with V is:
W1 � ��U ���� V�
Υ0 Ω0 � e1 Υ1 Ω0 � e2 Υ2 Ω0 � ��e1 � e1� Υ1 � �e1 � e2� Υ2� Ω1 �
�e1 � e2 � e1� Υ3 Ω1 � ��e1 � e2� Υ1 � �e2 � e2� Υ2� Ω2 �
�e1 � e2 � e2� Υ3 Ω2 � �e1 � e2 � e1 � e2� Υ3 Ω3 �Υ3 Ω0 e1 � e2
Note that there are no products of the form ei ���� �ej � ek� as these have been put to zero by
GrassmannSimplify. If we wish, we can convert the remaining interior products to scalar
products using the GrassmannAlgebra function ToScalarProducts.
W2 � ToScalarProducts�W1�
Υ0 Ω0 � e1 Υ1 Ω0 � e2 Υ2 Ω0 � �e1 � e1� Υ1 Ω1 � �e1 � e2� Υ2 Ω1 �
�e1 � e2� e1 Υ3 Ω1 � �e1 � e1� e2 Υ3 Ω1 � �e1 � e2� Υ1 Ω2 �
�e2 � e2� Υ2 Ω2 � �e2 � e2� e1 Υ3 Ω2 � �e1 � e2� e2 Υ3 Ω2 �
�e1 � e2�2 Υ3 Ω3 � �e1 � e1� �e2 � e2� Υ3 Ω3 �Υ3 Ω0 e1 � e2
Finally, we can substitute the values from the currently declared metric tensor for the scalar
products using the GrassmannAlgebra function ToMetricForm. For example, if we first
declare a general metric:
DeclareMetric���
���1,1, �1,2�, ��1,2, �2,2��
ToMetricForm�W2�
Υ0 Ω0 � e1 Υ1 Ω0 � e2 Υ2 Ω0 �Υ1 Ω1 �1,1 � e2 Υ3 Ω1 �1,1 �
2
Υ2 Ω1 �1,2 � e1 Υ3 Ω1 �1,2 �Υ1 Ω2 �1,2 � e2 Υ3 Ω2 �1,2 �Υ3 Ω3 �1,2 Υ2 Ω2 �2,2 � e1 Υ3 Ω2 �2,2 �Υ3 Ω3 �1,1 �2,2 �Υ3 Ω0 e1 � e2
The interior product of two general Grassmann numbers in 3-space is
2001 4 5
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